L11n283

L11n282

L11n284

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n283 at Knotilus!

Link Presentations

[edit Notes on L11n283's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X19,10,20,5
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -3, -6, 11}, {-8, -2, 4, -5, 10, 9, -7, 6, -11, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{2}(w-1)^{2}}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+6q^{4}-9q^{3}-q^{-3}+11q^{2}+5q^{-2}-10q-6q^{-1}+11$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+z^{2}a^{-4}-a^{-4}z^{-2}-a^{-4}-z^{6}a^{-2}-3z^{4}a^{-2}-a^{2}z^{2}-2z^{2}a^{-2}+2a^{2}z^{-2}+4a^{-2}z^{-2}+a^{2}+4a^{-2}+2z^{4}+2z^{2}-5z^{-2}-4$ (db)
Kauffman polynomial	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+2az^{7}+2z^{7}a^{-3}+4z^{7}a^{-5}-28z^{6}a^{-2}-15z^{6}a^{-4}+z^{6}a^{-6}-12z^{6}-az^{5}-8z^{5}a^{-1}-19z^{5}a^{-3}-12z^{5}a^{-5}+5a^{2}z^{4}+29z^{4}a^{-2}+11z^{4}a^{-4}-2z^{4}a^{-6}+21z^{4}+a^{3}z^{3}+az^{3}+6z^{3}a^{-1}+13z^{3}a^{-3}+7z^{3}a^{-5}-3a^{2}z^{2}-12z^{2}a^{-2}-4z^{2}a^{-4}-11z^{2}+5az+9za^{-1}+5za^{-3}+za^{-5}-3a^{2}-2a^{-2}-4-5az^{-1}-9a^{-1}z^{-1}-5a^{-3}z^{-1}-a^{-5}z^{-1}+2a^{2}z^{-2}+4a^{-2}z^{-2}+a^{-4}z^{-2}+5z^{-2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 χ 13                   1 1 11                 3   -3 9               3 1   2 7             6 3     -3 5           5 3       2 3         5 6         1 1       6 5           1 -1     2 7             5 -3   3 4               -1 -5   4                 4 -7 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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